SAR is a signal processing method that can be used to obtain images from an aircraft, or more generally some kind of a platform carrying radar equipment, of the ground with a resolution approaching the resolution of optical systems. Actually a resolution down to the order of half the radar wavelength is possible. Such a resolution is achieved by the radar imaging the ground continuously within some given straight segment of a length L of the platform or aircraft path. The attained angular, also called the azimuth, ground resolution measured along a circular arc at the distance R from the platform will then be:
      δ    ⁢                  ⁢    θ    =            2      ⁢      c                      (                              F            max                    +                      F            min                          )            ⁢                        tan                      -            1                          ⁡                  (                                    L              /              2                        ⁢            R                    )                    where C is the speed of light and Fmax, Fmin are the upper and lower limits of the frequency band used by the radar. This formula can be so interpreted that the possible solution is inversely proportional to total aspect angle variation Δθ=tan−1(L/2R) occurring during the imaging process. In most SAR systems the interval L is small compared to the distance R between the aircraft path and the imaged objects which means that Δθ will be small. Thus only a ground resolution which is much larger than the radar wavelength at the mean frequency
      λ    _    =            2      ⁢      c                      F        max            +              F        min            can be attained. It is desirable to be able to improve the resolution and therefore attempts have been done with much larger aspect angle variation. As this angle approaches its limit of 180°, ground resolution approaches its theoretical limit of λ/π. Diffraction limited (DL) SAR imaging means SAR imaging attaining wavelength order ground resolution. However, approaching the diffraction resolution limit involves a number of signal processing problems that need to be solved.
Even if L/ λ is large for DL systems, the physical size of the antenna of the radar system is generally not increased. For so called strip map systems, it must be of the same order of extension as λ. This means that for DL SAR the computational effort becomes large per unit surveyed ground area.
Normally computer efficient processing methods are based on plane wave approximations of radar raw data, whereas a large L/R requires a spherical wave representation of radar raw data. That makes the SAR processing task much more difficult.
Small deviations from a straight aircraft track must be compensated for and this can be done through an accurate navigation with information about the deviations, and compensating the signal processing for such known deviations. Alternatively compensation can be achieved by implementing so called autofocus in which the processing itself involves the task of removing the imaging errors due to a non-linear aircraft path. There are efficient methods for performing compensation for both cases, i.e. an accurate navigation with known deviations or by using autofocus. However it is a drawback that the methods only can be used for plane wave approximations of radar raw data.
DL imaging is implemented in VHF SAR. In the CARABAS™ system which is a Swedish system, Fmax≈85 MHz and Fmin≈25 MHz whereas Δθ≈60°. The ground resolution will then be of the order of magnitude 2 meters. Less extreme but also close to the diffraction limit is an X-band SAR system attaining a ground resolution of 0.1 m.
DL SAR processing methods exist which can be used when the SAR path, i.e. a path of the platform or the aircraft, is accurately known. In order to make the systems of practical use without putting to high requirements on navigation and making the systems too expensive, these methods ought to be generalized so that they can be applied also when knowledge of the platform path is lacking or is less precise.
The principle of non-DL motion error compensation is explained in FIG. 1. Non-DL SAR imaging used in microwave SAR is based on the assumption that radar waves are approximately plane across the imaged area. Then motion errors affect the SAR raw data by range translations. SAR focusing can thus be achieved just by range adjustments in the raw data as can be seen in FIG. 1 wherein A, B indicate an ideal straight SAR path, whereas the curved line CD indicates the actually flow path, assuming radar wavefronts across the image to be approximately plane, actual data collected at P and at certain range (i.e. along the intersection between ground and the radar wavefront W1) will thereby be approximately identical to ideal data collected at Q along a straight path at another range (i.e. along the intersection between the ground and the radar wavefront W2). Thus by introducing appropriate range shifts in the data, these data can be attributed to the correct ground points, whilst the straight path assumption kept.
Considering DL SAR processing, the most obvious way for performing DL SAR processing is based on use of a range migration algorithm, RMA, which can be implemented to be computationally fast by means of FFTs (Fast Fourier Transform). It is similar to Fourier based approximate methods of non-DL SAR processing.
RMA recognizes that the radar waves will be spherical across the imaged area. RMA however crucially depends on that, for a platform moving along a straight line, such spherical waves can be transformed into a plane wave expansion, and the SAR processing cast in a form similar to non-DL imaging.
If the spherical nature of the radar waves is to be taken into account, track deviations cannot be represented as range shifts, which is illustrated in FIG. 2, wherein the same reference figures are used as in FIG. 1. DL imaging adopting RMA assumes spherical radar wavefronts, intersecting the (roughly) plane ground in circles. Again a SAR platform has attempted to follow an ideal straight path AB, whereas CD is the actual path flown. The situation is thus that data collected at P and at a certain range (i.e. along the intersection between ground and the radar wavefront W1) no longer will be identical to ideal data collected at any point Q along a straight for any other range. It is no longer trivial to transform data collected along the actual path to fit a straight path assumption. The possibility to compensate data by equal data that would have been captured along a straight track is hence lost and the RMA method will not be applicable. Instead of RMA, for performing DL SAR processing, so called global backprojection, GBP, may be used. This is a technique that also is used in computer tomography. It is however a drawback of GBP that it is not computationally efficient. Therefore the areas to be imaged have to be quite small. An advantage of GBP is however that it does not involve any assumption concerning the straightness of the platform path. Thus, if the platform moves in a known but non-straight manner, GBP can be used for SAR processing.
However, when the platform path deviates from a straight line, also the ground topography becomes of importance for focusing. This means that in such cases also the ground topography has to be known albeit the accuracy does not have to be very high if the deviations from a straight path are small.
Motion compensation by range shifts in non-DL SAR imaging does not require full knowledge of the SAR path. However, DL SAR focusing requires full knowledge of the SAR path. Since L/ λ is large the SAR path will involve many degrees of freedom all of which must be made known with wavelength dependent accuracy, which is very complicated and puts very high requirements on equipment, processing means etc.
For non-DL SAR, and since L/ λ is small, dominant ground reflectors will be apparent already in raw data and can be used to estimate the range shifts caused by the deviations from a straight track platform path. For strip-map DL imaging, the physical antenna of the radar system must be small in relation to λ, i.e. in wavelength units even smaller than for non-DL SAR imaging. Such a small antenna does not provide any initial resolution which means that for all except for very unusual types of ground, for example large industrial plants, there will be no dominant ground reflectors apparent in the raw data which is an extra complication in DL imaging. Due to the above discussed characteristics of DL imaging, it is apparent that DL autofocus is a most complicated issue.
In order to be able to better handle motion errors, a number of so called local backprojection methods, LBP, have been developed. These have the same capability to take into account known motion errors in DL SAR as GBP. However, they are numerically much more efficient and by using LBP, it gets possible to obtain real or near real time processing of for example CARABAS data with significant aerial coverage.
One such method is the so called Factorized Fast Backprojection, FFB, as also described in “Synthetic-Aperture Radar Processing using Fast Factorised Back-Projection, IEEE Trans. Aerospace and Electronic Systems, Vol. 39, No. 3, pp. 760-776, 2003 by Ulander, L., Hellsten, H. and Stenstrom, G. A base n FFB SAR-processing algorithm produces a SAR image based on a raw data set consisting of radar range returns from nk position, where n, and k are integers, distributed over a platform path segment of length L. Typically n=2 or n=3 whereas k≈10. However, for error growth reducing purposes, also n≈10 with k≈8 can be considered. The FFB SAR image reconstruction occurs in k iterations whereby each iteration performs a subaperture coalescing, forming one new subaperture for every n neighboring subaperture defined in the previous iteration. To every subaperture at every level of iteration there is associated a SAR image of the same ground portion. Coalescing subapertures can, based on geometrical data for them and their orientation with respect to the ground, be associated with a linear combination of the associated ground images upon which the new subaperture becomes associated with a single new SAR image with angular resolution improved by the factor n.
The subapertures of the first iteration are defined to have lengths equal to the separation between the nk data positions. This separation is generally some fraction of the real aperture of the radar system. The associated SAR images of the first iteration are simply the nk range returns, each only possessing the angular resolution of the real aperture of the radar system.
The advantage of FFB is that, since angular resolution increases exponentially with each iteration, the image representations at initial iterations allow a coarse level of discretization saving computational effort. Only the last iteration requires a full discretization of the final SAR image. In fact, for a N×N point SAR image, where there are N=nk data positions along the SAR path, the FFB computational effort is of the order N2×n log N. This means that the computational effort is comparable to that of RMA as well as of Fourier based methods of non-DL SAR, which require a processing effort of the order N2×2 log N.